A complete Computer Algebra System
Maxima is a CAS (Computer Algebra System), similar to systems like Mathematica and Maple, designed for the manipulation of algebraic expressions. But you can use Maxima for manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, Ordinary Differential Equations (ODE), systems of linear equations, polynomials, and sets, lists, vectors, matrices, and tensors.
It can differentiate, integrate, take limits, solve equations, factor polynomials, expand functions in power series, solve differential equations in closed form, and perform many other operations. Maxima also can handle trigonometric functions, but there are lots of identities you can use to help simplify equations with trig functions in them. It can plot functions and data in two and three dimensions.
Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. It also has a programming language that you can use to extend Maxima’s capabilities (see, and/or download, below the complete manual!).
Maxima, however, is a command line application which makes it a bit harder to use. Here is where a GUI (Graphical User Interface) for Maxima made to make using Maxima simpler and more enjoyable.
wxMaxima - A cross platform GUI
wxMaxima is a graphical user interface (GUI) for the computer algebra system Maxima. It eases the use of Maxima by making most of its commands available through a menu system and by providing input dialogs for commands that require more than one argument. It also implements its own display engine that outputs mathematical symbols directly instead of depicting them with ASCII characters. wxMaxima also features 2D and 3D inline plots, simple animations, mixing of text and mathematical calculations to create documents, exporting of input and output to TeX, and a browser for Maxima's manual including command index and full text searching. wxMaxima uses wxWidgets.